Properties

Label 25215.f
Number of curves $8$
Conductor $25215$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 25215.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
25215.f1 25215h8 \([1, 0, 0, -3630995, -2663397180]\) \(1114544804970241/405\) \(1923792217605\) \([2]\) \(266240\) \(2.1477\)  
25215.f2 25215h6 \([1, 0, 0, -226970, -41617125]\) \(272223782641/164025\) \(779135848130025\) \([2, 2]\) \(133120\) \(1.8011\)  
25215.f3 25215h7 \([1, 0, 0, -184945, -57494170]\) \(-147281603041/215233605\) \(-1022382059916218805\) \([2]\) \(266240\) \(2.1477\)  
25215.f4 25215h4 \([1, 0, 0, -134515, 18977882]\) \(56667352321/15\) \(71251563615\) \([2]\) \(66560\) \(1.4545\)  
25215.f5 25215h3 \([1, 0, 0, -16845, -390600]\) \(111284641/50625\) \(240474027200625\) \([2, 2]\) \(66560\) \(1.4545\)  
25215.f6 25215h2 \([1, 0, 0, -8440, 293567]\) \(13997521/225\) \(1068773454225\) \([2, 2]\) \(33280\) \(1.1079\)  
25215.f7 25215h1 \([1, 0, 0, -35, 12840]\) \(-1/15\) \(-71251563615\) \([2]\) \(16640\) \(0.76136\) \(\Gamma_0(N)\)-optimal
25215.f8 25215h5 \([1, 0, 0, 58800, -2917143]\) \(4733169839/3515625\) \(-16699585222265625\) \([2]\) \(133120\) \(1.8011\)  

Rank

sage: E.rank()
 

The elliptic curves in class 25215.f have rank \(1\).

Complex multiplication

The elliptic curves in class 25215.f do not have complex multiplication.

Modular form 25215.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} + q^{15} - q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.